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Chapter 4: Problem 8

Using the Pauli matrix representation, reduce each of the operators (a) \(s_{x} s_{y}\) (b) \(s_{x} s_{y}^{2} s_{z}^{2},\) and (c) \(s_{x}^{2} s_{y}^{2} s_{z}^{2},\) to a single spin operator.

Short Answer

Expert verified
(a) The operator \(s_{x} s_{y}\) reduces to \(\frac{1}{2i}s_{z}\). (b) The operator \(s_{x} s_{y}^{2} s_{z}^{2}\) reduces to \(s_{x}\). (c) The operator \(s_{x}^{2} s_{y}^{2} s_{z}^{2}\) reduces to I.

Step by step solution

01

Reduce the operator \(s_{x} s_{y}\)

Use the property that \(s_{x}s_{y} = \frac{1}{2i}s_{z}\). This means the operator \(s_{x}s_{y}\) reduces to a single spin operator \(\frac{1}{2i}s_{z}\).
02

Reduce the operator \(s_{x} s_{y}^{2} s_{z}^{2}\)

Considering the given operator \(s_{x} s_{y}^{2} s_{z}^{2}\), note that since \(s_{y}^{2} = I\) and \(s_{z}^{2} = I\), then \(s_{y}^{2} s_{z}^{2} = I\). Therefore \(s_{x} s_{y}^{2} s_{z}^{2}\) simplifies to \(s_{x} I\) which equals \(s_{x}\).
03

Reduce the operator \(s_{x}^{2} s_{y}^{2} s_{z}^{2}\)

The given operator \(s_{x}^{2} s_{y}^{2} s_{z}^{2}\) reduces to \(I\), because \(s_{x}^{2} = I\), \(s_{y}^{2} = I\), and \(s_{z}^{2} = I\). Therefore: \(s_{x}^{2} s_{y}^{2} s_{z}^{2} = I\).

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Most popular questions from this chapter

Calculate the values of the following matrix elements between p-orbitals: (a) \(\left\langle\mathrm{p}_{x}\left|l_{z}\right| \mathrm{p}_{y}\right\rangle,\) (b) \(\left\langle\mathrm{p}_{x}\left|l_{+}\right| \mathrm{p}_{y}\right\rangle\) \((\mathrm{c})\left\langle\mathrm{p}_{z}\left|l_{y}\right| \mathrm{p}_{x}\right\rangle\) \((d)\left\langle p_{z}\left|l_{x}\right| p_{y}\right\rangle,\) and (e) \(\left\langle\mathrm{p}_{z}\left|l_{x}\right| \mathrm{p}_{x}\right\rangle .\) Hint. Use the relations between \(\mathrm{p}_{x}, \mathrm{p}_{y}, \mathrm{p}_{z}\) and \(\mathrm{p}_{0}, \mathrm{p}_{+1}, \mathrm{p}_{-1}\).

Suppose that in place of the actual angular momentum commutation rules, the operators obeyed \(\left[l_{x}, l_{y}\right]=-\mathrm{i} \hbar l_{z^{*}}\) What would be the roles of \(l_{2}=l_{x} \pm l_{y} ?\)

(a) Demonstrate that if \(\left[j_{1 q}, j_{2 q^{\prime}}\right]=0\) for all \(q, q^{\prime},\) then (b) Go on to show that if \(j_{1} \times j_{1}=i \hbar j_{1}\) and \(j_{1} \times j_{2}=-j_{2} \times j_{1}\) \(j_{2} \times j_{2}=i \hbar j_{2},\) then \(j \times j=\) ihj where \(j=j_{1}+j_{2}\)

Determine what total angular momenta may arise in the following composite systems: (a) \(j_{1}=3, j_{2}=4 ;\) (b) the orbital momenta of two electrons (i) both in p-orbitals, (ii) both in d-orbitals, (iii) the configuration \(\mathrm{p}^{1} \mathrm{d}^{1} ;\) (c) the spin angular momenta of four electrons. Hint. Use the ClebschGordan series, eqn \(4.42 ;\) apply it successively in (c).

What are the possible outcomes of a single measurement of the \(z\) -component of spin angular momentum of an electron in the spin state \((1 / 4)^{1 / 2} \alpha-(3 / 4)^{1 / 2} \beta ?\)
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